Question

fluid dynamics pipe flow problem with solution

Answer 1

Certainly! Here’s a classic fluid dynamics problem involving flow in a pipe along with its solution.

### Problem Statement
Consider water (aqueduct) flowing through a horizontal pipe with a diameter of 0.1 m (10 cm) and with a flow rate of 0.02 m³/s. The pipe is smooth, and we would like to determine:

1. The flow velocity in the pipe.
2. The Reynolds number to classify the flow regime.
3. The head loss due to friction over a length of 50 m.

### Given Data:
- Diameter of the pipe, \( D = 0.1 \) m
- Flow rate, \( Q = 0.02 \) m³/s
- Length of the pipe, \( L = 50 \) m
- Density of water, \( \rho \approx 1000 \) kg/m³
- Dynamic viscosity of water, \( \mu \approx 0.001 \) Pa·s

### Solution Steps:

1. **Flow Velocity Calculation:**
The flow velocity \( V \) can be calculated using the flow rate \( Q \) and the cross-sectional area \( A \) of the pipe.

\[
A = \frac{\pi D^2}{4} = \frac{\pi (0.1)^2}{4} \approx 0.00785 \, \text{m}^2
\]

Now, calculate the velocity \( V \):

\[
V = \frac{Q}{A} = \frac{0.02}{0.00785} \approx 2.55 \, \text{m/s}
\]

2. **Reynolds Number Calculation:**
The Reynolds number \( Re \) is given by:

\[
Re = \frac{\rho V D}{\mu}
\]

Plugging in the values:

\[
Re = \frac{1000 \times 2.55 \times 0.1}{0.001} = 255000
\]

Since \( Re > 4000 \), the flow is classified as turbulent.

3. **Head Loss Calculation Using Darcy-Weisbach Equation:**
The head loss \( h_f \) due to friction can be calculated using the Darcy-Weisbach equation:

\[
h_f = f \frac{L}{D} \frac{V^2}{2g}
\]

We first need to find the Darcy friction factor \( f \). For turbulent flow in smooth pipes, we can use the Colebrook-White equation or approximate using the Moody chart.

For a smooth pipe, a common approximation is:

\[
f \approx 0.3164 Re^{-0.25}
\]

Calculating \( f \):

\[
f \approx 0.3164 \times (255000)^{-0.25} \approx 0.0141
\]

Now we can substitute \( f \), \( L \), \( D \), \( V \), and \( g \) (acceleration due to gravity, approximately 9.81 m/s²) into the Darcy-Weisbach equation:

\[
h_f = 0.0141 \times \frac{50}{0.1} \times \frac{(2.55)^2}{2 \times 9.81}
\]

\[
h_f = 0.0141 \times 500 \times \frac{6.5025}{19.62} \approx 0.0141 \times 500 \times 0.331 \approx 2.33 \, \text{m}
\]

### Summary of Results:
1. Flow velocity, \( V \approx 2.55 \, \text{m/s} \)
2. Reynolds number, \( Re \approx 255000 \) (turbulent flow)
3. Head loss due to friction over the length of the pipe, \( h_f \approx 2.33 \, \text{m} \)

This example illustrates the steps involved in solving a simple fluid dynamics problem regarding flow in a pipe.